In Fever 1793 by Laurie Halse Anderson, why does Benjamin Rush enlist the help of the Free African Society to treat the fever victims?

It tells you something about the primitive state of medicine at the time that it was genuinely believed, by a qualified medical practitioner, that African Americans were immune from yellow fever. Yet that it precisely what Dr. Rush believes. His understanding—or lack thereof—of this terrible condition forces him to take desperate measures. For white people to enlist the help of African-Americans at that time would've been considered quite desperate indeed.
In addition to scientific ignorance, Dr. Rush is motivated by racism. He sees African Americans as strange and mysterious, not fully human by comparison with white folk. This is the main factor in his absurd conclusion that black people are immune from yellow fever and can therefore be called upon to help treat yellow fever victims and bury the dead.

Dr. Benjamin Rush, who led the medical fight against the yellow fever in Philadelphia in 1793, asked free Africans in the city to help treat the fever victims because he believed that they were immune to the disease. In the novel, Eliza tells Mattie that Dr. Rush wrote to Reverend Allen from the Free Africa Society asking his members to provide help for the fever victims. Eliza explains that Reverend Allen felt that it was an opportunity for African-American people to show that they merited equality with white people. 
In the actual epidemic, African-American people cared for the sick and buried them. After the epidemic, Reverend Allen founded the African Methodist Episcopal Church, an African-American church, in Philadelphia. Dr. Rush's method of treating patients was to leech their blood and purge them, a practice that was attacked by other doctors and journalists. There is still no cure for yellow fever, though a vaccine to prevent the disease was developed in 1937.